There are 1600 students at a school. Each student has either a packed lunch or a school dinner. 62% of the students have school dinners. 55% of the students are girls. 45% of the girls have packed lunches. How many boys have school dinners? You must show all your working.

There are 1600 students, and 62% have school dinners.

$$\text{Students with school dinners} = 0.62 \times 1600 = 992$$

55% of the students are girls.

$$\text{Number of girls} = 0.55 \times 1600 = 880$$

45% of the girls have packed lunches.

$$\text{Girls with packed lunches} = 0.45 \times 880 = 396$$

Since each student has either a packed lunch or a school dinner, the remaining girls must have school dinners.

$$\text{Girls with school dinners} = \text{Total girls} - \text{Girls with packed lunches}$$ $$\text{Girls with school dinners} = 880 - 396 = 484$$

The total number of students with school dinners is 992 (from Step 1). If 484 of these are girls, the rest must be boys.

$$\text{Boys with school dinners} = \text{Total students with school dinners} - \text{Girls with school dinners}$$ $$\text{Boys with school dinners} = 992 - 484 = 508$$

There are 508 boys who have school dinners.

The final answer is 508.